Umbilical cord tissue is a rich source of stem cells. Blood from the umbilical cord includes stem cells, including hematopoietic stem cells that can be used to repopulate a person's blood and immune system. Wharton's Jelly, a gelatinous substance within the umbilical cord, contains an additional population of stem cells, distinct from those found in cord blood. As used herein, “Wharton's Jelly” can further include the amniotic epithelial layer of the umbilical cord. Processing and culturing the Wharton's Jelly permits the isolation of mesenchymal stem cells that can be used to regenerate a variety of tissues (see, for example, U.S. Pat. No. 5,919,702).